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440 lines
9.3 KiB
Markdown
440 lines
9.3 KiB
Markdown
section 2eW, lite number theory
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===============================
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### ++egcd
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GCD
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++ egcd :: schneier's egcd
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|= [a=@ b=@]
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=+ si
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=+ [c=(sun a) d=(sun b)]
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=+ [u=[c=(sun 1) d=--0] v=[c=--0 d=(sun 1)]]
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|- ^- [d=@ u=@ v=@]
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?: =(--0 c)
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[(abs d) d.u d.v]
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:: ?> ?& =(c (sum (pro (sun a) c.u) (pro (sun b) c.v)))
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:: =(d (sum (pro (sun a) d.u) (pro (sun b) d.v)))
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:: ==
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=+ q=(fra d c)
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%= $
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c (dif d (pro q c))
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d c
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u [(dif d.u (pro q c.u)) c.u]
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v [(dif d.v (pro q c.v)) c.v]
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==
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::
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Greatest common denominator
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~zod/try=> (egcd 20 15)
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[d=5 u=2 v=1]
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~zod/try=> (egcd 24 16)
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[d=8 u=2 v=1]
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~zod/try=> (egcd 7 5)
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[d=1 u=3 v=6]
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~zod/try=> (egcd (shaf ~ %ham) (shaf ~ %sam))
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[ d=1
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u=59.983.396.314.566.203.239.184.568.129.921.874.787
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v=38.716.650.351.034.402.960.165.718.823.532.275.722
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]
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------------------------------------------------------------------------
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### ++pram
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Probable prime
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++ pram :: rabin-miller
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|= a=@ ^- ?
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?: ?| =(0 (end 0 1 a))
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=(1 a)
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=+ b=1
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|- ^- ?
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?: =(512 b)
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?|(=+(c=+((mul 2 b)) &(!=(a c) =(a (mul c (div a c))))) $(b +(b)))
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==
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=+ ^= b
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=+ [s=(dec a) t=0]
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|- ^- [s=@ t=@]
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?: =(0 (end 0 1 s))
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$(s (rsh 0 1 s), t +(t))
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[s t]
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?> =((mul s.b (bex t.b)) (dec a))
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=+ c=0
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|- ^- ?
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?: =(c 64)
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&
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=+ d=(~(raw og (add c a)) (met 0 a))
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=+ e=(~(exp fo a) s.b d)
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?& ?| =(1 e)
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=+ f=0
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|- ^- ?
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?: =(e (dec a))
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&
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?: =(f (dec t.b))
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$(e (~(pro fo a) e e), f +(f))
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==
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$(c +(c))
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==
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::
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Probable prime test
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~zod/try=> (pram 31)
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%.y
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~zod/try=> =+(a=2 |-(?:(=(a 31) ~ [i=(mod 31 a) t=$(a +(a))])))
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~[1 1 3 1 1 3 7 4 1 9 7 5 3 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1]
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~zod/try=> =+(a=2 |-(?:(=(a 31) ~ [i=(mod 30 a) t=$(a +(a))])))
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~[0 0 2 0 0 2 6 3 0 8 6 4 2 0 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0]
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~zod/try=> (pram 256)
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%.n
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~zod/try=> (pram (dec (bex 127)))
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%.y
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------------------------------------------------------------------------
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### ++ramp
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`r-m` prime
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++ ramp :: make r-m prime
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|= [a=@ b=(list ,@) c=@] ^- @ux :: [bits snags seed]
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=> .(c (shas %ramp c))
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=+ d=_@
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|-
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?: =((mul 100 a) d)
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~|(%ar-ramp !!)
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=+ e=(~(raw og c) a)
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?: &((levy b |=(f=@ !=(1 (mod e f)))) (pram e))
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e
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$(c +(c), d (shax d))
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::
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Random `a` bit prime, which isn't 1 modulo a list of other numbers,
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using salt `c`.
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~zod/try=> (ramp 20 ~ %hamelok)
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0xf.1f0d
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~zod/try=> (ramp 20 ~ %hameloe)
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0x2.d341
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~zod/try=> (ramp 5 ~ %kole)
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0x1f
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~zod/try=> (ramp 7 ~ %kole)
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0x4f
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~zod/try=> (ramp 7 ~[0x4e] %kole)
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0x43
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~zod/try=> `@uw`(ramp 128 ~ %late)
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0w3y.irKIL.l-pp1.2CkG4.3lsTF
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------------------------------------------------------------------------
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### ++fo
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Prime engine
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++ fo :: modulo prime
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|_ a=@
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XX DO NOT RERUN GET.LS, THERE EXIST ARM COLLISIONS
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Core for performing arithmetic modulo a prime number
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~zod/try=> ~(. fo 79)
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<7.get [@ud <373.jdd 100.kzl 1.ypj %164>]>
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------------------------------------------------------------------------
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### ++dif
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Difference
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++ dif
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|= [b=@ c=@]
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(sit (sub (add a b) (sit c)))
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::
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Subtract
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~zod/try=> (~(dif fo 79) 10 5)
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5
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~zod/try=> (~(dif fo 79) 5 10)
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74
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------------------------------------------------------------------------
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### ++exp
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Exponent
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++ exp
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|= [b=@ c=@]
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?: =(0 b)
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1
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=+ d=$(b (rsh 0 1 b))
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=+ e=(pro d d)
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?:(=(0 (end 0 1 b)) e (pro c e))
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::
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Exponent
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~zod/try=> (~(exp fo 79) 3 5)
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46
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------------------------------------------------------------------------
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### ++fra
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Divide
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++ fra
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|= [b=@ c=@]
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(pro b (inv c))
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::
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Divide
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~zod/try=> (~(fra fo 79) 20 4)
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5
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~zod/try=> (~(fra fo 79) 7 11)
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15
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------------------------------------------------------------------------
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### ++inv
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Inverse
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++ inv
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|= b=@
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=+ c=(dul:si u:(egcd b a) a)
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c
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::
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Multiplicative inverse
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~zod/try=> (~(inv fo 79) 12)
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33
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~zod/try=> (~(pro fo 79) 12 33)
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1
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~zod/try=> (~(inv fo 79) 0)
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0
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------------------------------------------------------------------------
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### ++pro
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Product
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++ pro
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|= [b=@ c=@]
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(sit (mul b c))
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::
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Product
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~zod/try=> (~(pro fo 79) 5 10)
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50
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~zod/try=> (~(pro fo 79) 5 20)
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21
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------------------------------------------------------------------------
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### ++sit
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Bounds
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++ sit
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|= b=@
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(mod b a)
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::
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Bounds check
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~zod/try=> (~(sit fo 79) 9)
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9
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~zod/try=> (~(sit fo 79) 99)
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20
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------------------------------------------------------------------------
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### ++sum
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Sum
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++ sum
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|= [b=@ c=@]
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(sit (add b c))
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--
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Add
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~zod/try=> (~(sum fo 79) 9 9)
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18
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~zod/try=> (~(sum fo 79) 70 9)
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0
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------------------------------------------------------------------------
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### ++ga
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++ ga :: GF (bex p.a)
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|= a=[p=@ q=@ r=@] :: dim poly gen
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=+ si=(bex p.a)
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=+ ma=(dec si)
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=> |%
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RSA internals
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XX document
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------------------------------------------------------------------------
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### ++dif
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++ dif :: add and sub
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|= [b=@ c=@]
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~| [%dif-ga a]
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?> &((lth b si) (lth c si))
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(mix b c)
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::
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XX document
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------------------------------------------------------------------------
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### ++dub
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++ dub :: mul by x
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|= b=@
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~| [%dub-ga a]
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?> (lth b si)
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?: =(1 (cut 0 [(dec p.a) 1] b))
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(dif (sit q.a) (sit (lsh 0 1 b)))
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(lsh 0 1 b)
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::
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XX document
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------------------------------------------------------------------------
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### ++pro
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++ pro :: slow multiply
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|= [b=@ c=@]
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?: =(0 b)
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0
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?: =(1 (dis 1 b))
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(dif c $(b (rsh 0 1 b), c (dub c)))
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$(b (rsh 0 1 b), c (dub c))
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::
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XX document
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------------------------------------------------------------------------
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### ++toe
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++ toe :: exp/log tables
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=+ ^= nu
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|= [b=@ c=@]
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^- (map ,@ ,@)
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=+ d=*(map ,@ ,@)
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|-
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?: =(0 c)
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d
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%= $
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c (dec c)
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d (~(put by d) c b)
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==
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=+ [p=(nu 0 (bex p.a)) q=(nu ma ma)]
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=+ [b=1 c=0]
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|- ^- [p=(map ,@ ,@) q=(map ,@ ,@)]
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?: =(ma c)
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[(~(put by p) c b) q]
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%= $
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b (pro r.a b)
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c +(c)
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p (~(put by p) c b)
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q (~(put by q) b c)
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==
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::
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XX document
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------------------------------------------------------------------------
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### ++sit
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++ sit :: reduce
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|= b=@
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(mod b (bex p.a))
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--
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XX document
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------------------------------------------------------------------------
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### ++fra
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++ fra :: divide
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|= [b=@ c=@]
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(pro b (inv c))
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::
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XX document
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------------------------------------------------------------------------
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### ++inv
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++ inv :: invert
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|= b=@
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~| [%inv-ga a]
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=+ c=(~(get by q) b)
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?~ c !!
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=+ d=(~(get by p) (sub ma u.c))
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(need d)
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::
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XX document
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------------------------------------------------------------------------
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### ++pow
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++ pow :: exponent
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|= [b=@ c=@]
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=+ [d=1 e=c f=0]
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|-
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?: =(p.a f)
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d
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?: =(1 (cut 0 [f 1] b))
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$(d (pro d e), e (pro e e), f +(f))
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$(e (pro e e), f +(f))
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::
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XX document
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------------------------------------------------------------------------
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### ++pro
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++ pro :: multiply
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|= [b=@ c=@]
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~| [%pro-ga a]
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=+ d=(~(get by q) b)
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?~ d 0
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=+ e=(~(get by q) c)
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?~ e 0
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=+ f=(~(get by p) (mod (add u.d u.e) ma))
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(need f)
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--
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XX document
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------------------------------------------------------------------------
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